| 摘要: |
| 图的临界群是一个Abelian群，其不变因子与图的Laplacian矩阵、圈空间和键空间密切相关. 本文主要研究完全图$K_{3}$与路$P_{n}$的直积（记为$K_{3}\bullet P_{n}$）的临界群结构，得到$K_{3}\bullet P_{n} (n\geq 3)$的临界群恒为三个循环群的直和分解. 这一研究为深入研究其它乘积图的的临界群结构提供了参考. |
| 关键词: 临界群 Smith群 Laplacian矩阵 图的直积 |
| DOI： |
| 分类号:O157.5 |
| 基金项目:国家自然科学 基金资助(12161073) |
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| The critical group structure of the direct product of path and $K_ {3}$ |
|
Ren Haizhen
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Qinghai Normal University
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| Abstract: |
| The critical group of a graph is an Abelian group, whose invariant factors are closely related to the Laplacian matrix, cycle space, and bond space of the graph. This article mainly studies the critical group structure of the direct product of the complete graph $K_{3} $and the path $P_ {n} $(denoted as $K_{3} \bullet P_ {n} $), showing that the critical group of $K_{3} \bullet P_ {n} (n \geq 3) $ always decomposes into the direct sum of three cyclic groups. This study provides a reference for further investigating the critical group structure of other product graphs. |
| Key words: Critical group Smith group Laplacian matrix Direct product of graphs |
